Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials

Transcription

1 Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials Shi Jin and Xin Wen Abstract When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of selecting a unique, physically relevant solution across the potential barrier, and the problem of a severe time step constraint due to the CFL condition. In this paper, We introduce two classes of Hamiltonian-preserving schemes for such problems. By using the constant Hamiltonian across the potential barrier, we introduced a selection criterion for a unique, physically relavant solution to the underlying linear hyperbolic equation with singular coefficients. These scheme have a hyperbolic CFL condition, which is a significant improvement over a conventional discretization. We also establish the positivity, and stability in both l and l norms, of these discretizations, and conducted numerical experiments to study the numerical accuracy. This work is motivated by the well-balanced kinetic schemes by Perthame and Simeoni for the shallow water equations with a discontinuous bottom topography, and has applications to the level set methods for the computations of multivalued physical observables in the semiclassical limit of the linear Schrödinger equation with a discontinuous potential, among other applications. Research supported in part by NSF grant No. DMS , NSFC under the Project 080 and the Basic Research Projects of Tsinghua University under the Project JC0000. Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA, and Department of Mathematical Sciences, Tsinghua University, Beijing 00084, P.R. China. address: jin@math.wisc.edu. Department of Mathematical Sciences, Tsinghua University, Beijing 00084, P.R. China. address: wenx@mail.tsinghua.edu.cn.

2 Introduction In this paper, we construct and study a class of numerical schemes for the d- dimensional Liouville equation in classical mechanics: f t + v x f x V v f = 0, t > 0, x,v R d, (.) where f(t,x,v) is the density distribution of a classical particle at position x, time t and traveling with velocity v. V (x) is the potential. The Liouville equation is a different formulation of Newton s second law: dx dt = v, which is a Hamiltonian system with the Hamiltonian dv dt = xv, (.) H = v + V (x) (.3) It is known from classical mechanics that the Hamiltonian remains constant across a potential barrier. The Liouville equation is a linear wave equation, with the characteristic speed determined by the Newton s equation (.) which is usually called the bicharacteristic. If V (x) is smooth, then the initial value problem to (.) is well-posed, and a standard numerical method (for example, the upwind scheme and its higher order extensions) for linear wave equation gives satisfactory results. However, if V (x) is discontinuous which corresponding to a potential barrier then the characteristic speed of the Liouville equation given by (.) is infinity at the discontinuous point. When numerically approximating V x across the interface, the numerical derivative of V is of O(/ x), with x the mesh size in the physical space. Thus an explicit scheme needs time step t = O( x ξ) with ξ the mesh size in particle velocity space. This is very expensive. Moreover, a conventional numerical scheme in general does not preserve a constant Hamiltonian across the interface, usually leads to poor or even incorrect numerical resolutions by ignoring the discontinuities of V (x). Theoretically, there is a uniqueness issue for weak solutions to these linear hyperbolic equations with singular wave speeds [, 4, 7, 0, ]. It is not clear which weak solution a standard numerical discretization that ignores the discontinuity of V (x) will select. Potential barriers appear in many important physical problems, such as the quantum tunneling (and quantum dots) in semiconductor device modeling, plasmas, and geometrical optics through different materials. Liouville or Vlasov equations describes the density distribution of particles in such a heterogeneous medium. For some recent mathematical study of discontinuous potentials in high frequency waves see [, 7, ]. In this paper, we construct a class of numerical schemes that are suitable for the Liouville equation (.) with a discontinuous potential. An important feature of

3 these schemes is that they are consistent to the constant Hamiltonian across a potential barrier for the Liouville equation (.). We call such schemes Hamiltonianpreserving schemes. A key idea in this paper is to use the behavior of a classical particle at the potential barrier either cross over if its kinetic energy is sufficiently large or be reflected with a negative velocity. We build this mechanics into the numerical scheme to construct the Hamiltonian-preserving schemes. This work is motivated by the well-balanced kinetic scheme of Perthame and Simeoni [9] for the shallow water equations with a (discontinuous) bottom topography, in which the similar mechanics was built into a hydrodynamic scheme for the shallow water equations in order to capture the steady state solutions corresponding to a constant energy of the shallow water equations. However, the work of Perthame and Simeoni was focused on a kinetic scheme for the shallow water equations defined in the physical space, thus the numerical discretization in the phase space was not studied. The phase space discretization is an important issue for the Liouville equation with a discontinuous potential. As shown by this work, if designed properly, the explicit Hamiltonian-preserving schemes allow a standard hyperbolic CFL condition t = O( x, ξ). More importantly, by using the a constant Hamiltonian condition across the potential barrier, these schemes select a unique, and physically relevant, solutions for the underlying linear hyperbolic equation with singular coefficients. Another application of the Liouville equation like (.) is the level set method for the computation of multivalued solutions to quasilinear PDEs, see [, 3]. Such problem arises in the semiclassical limit of the linear Schrödinger equation, which yields the Liouville equation (.) with the initial data f(x,v, 0) = ρ 0 (x)δ(v u 0 (x)), (.4) see for example [6, 9]. In the physical space, the moments of f: ρ(x, t) = f(x, v, t)dv, (.5) u(x, t) = f(x, v, t)vdv (.6) ρ(x, t) may become multivalued, see [0, 5] and the relevant numerical methods [8, 6]. The level set method proposed in [] solves the Liouville equation (.) with the initial data (.4) by decomposing f into φ and ψ i (i =,, d) where φ and ψ i solve the same Liouville equation (.) with initial data φ(x,v, 0) = ρ 0 (x), ψ i (x,v, 0) = v i u i0 (x), (.7) respectively. This allows the numerical computations for a bounded solution rather than measure-valued solution of the Liouville equation with singular initial data (.4), which greatly enhancing the numerical resolution. The moments can be recovered through ρ(x, t) = φ(x,v, t)π d i= δ(ψ i)dv, (.8) 3

4 u(x, t) = φ(x,v, t)vπ d i= δ(ψ i)dv/ρ(x, t) (.9) Numerical computations of multivalued solution for smooth potential using this technique were given in [] for smooth potentials. In this article we will also apply the Hamiltonian-preserving schemes for the level set computations of multivalued solution of the physical observables ρ,u, etc. In Sections, we first point out the problems with the usual finite difference scheme to solve the Liouville equation with discontinuous potentials. We then present the designing principle of our Hamiltonian-preserving schemes using the behavior of classical particles at a potential barrier. In Section 3, a d Hamiltonianpreserving scheme based on a finite difference approach (called Scheme I) is given, and its positivity and l are established. In Section 4, a D Hamiltonian-preserving scheme based on a finite volume approach (called Scheme II) is given. We extend these schemes to higher dimension in Section 5. In Section 6, we study the l - stability of Scheme I. We prove that the scheme is l -stable for suitable initial data, while for other (such as the measure-valued (.4)) initial data the solution may become unbounded. In Section 7, we prove that Scheme II is positive, l -stable, and l -contracting (for more general l initial data then Scheme I). In Section 8, we show that even for smooth initial data, the solution to the Liouville equation (.) could become discontinuous in the downstream part of the potential barrier. This contributes to the reduced numerical convergence rate to / for a formally first order scheme, as in any shock capturing method for a linear wave equation with discontinuous initial data. Numerical examples are given in Section 9 to verify the accuracy of the two schemes constructed in this paper. We conclude the paper in Section 0. The designing principle of the Hamiltonian-preserving schemes. Deficiency of the usual finite difference schemes We consider the numerical solution of the D Liouville equation f t + ξf x V x f ξ = 0 (.) with a discontinuous potential V (x). Without loss of generality, we employ an uniform mesh with grid points at x i+, i = 0,, N, in the x-direction and ξ j+, j = 0,, M in the ξ-direction. The cells are centered at (x i, ξ j ), i =,, N, j =,, M with x i = (x i+ + x i ) and ξ j = (ξ j+ + ξ j ). The mesh size is denoted by x = x i+ x i, ξ =. We also assume a uniform time step t and the discrete time is given by ξ j+ ξ j 4

5 0 = t 0 < t < < t L = T. We introduce mesh ratios λ t x = t, x λt ξ = t, ξ λξ x = ξ assumed to be fixed. We define the cell averages of f as f ij = x ξ xi+ ξj+ x i ξ j The -d average quantity f i+/,j is defined as f i+/,j = ξ ξj+ f(x, ξ, t)dξdx. ξ j f(x i+/, ξ, t)dξ. f,j+/ is defined similarly. A typical semi-discrete finite difference method for this equation is x, t f ij + ξ j f i+,j f i,j x f i,j+ f i,j DV i ξ = 0, (.) where the numerical fluxes f i+,j, f i,j+ are defined by the upwind scheme, and DV i is some numerical approximation of V x at x = x i. Such a discretization suffers from at least two problems: The above discretization in general does not preserve a constant Hamiltonian H = ξ + V across the discontinuities of V. Such a numerical approximation may lead to unphysical problem or poor numerical resolution. If an explicit time discretization is used, the CFL condition for this scheme requires the time step to satisfy [ maxj ξ j t + max ] i DV i. (.3) x ξ Since the potential V (x) is discontinuous at some points, max i DV i == O(/ x), so the CFL condition (.3) requires t = O( x ξ).. Behavior of a classical particle at a potential barrier In classical mechanics, a particle will either cross a potential barrier with a changing momentum, or be reflected, depending on its momentum and on the strength of the potential barrier. The Hamiltonian H = ξ + V should be preserved across the potential barrier: (ξ+ ) + V + = (ξ ) + V (.4) where the superscripts ± indicate the right and left limits of the quantity at the potential barrier. For example, consider the case when, at a potential discontinuity, the characteristic on the left of the potential discontinuity is given as a constant velocity ξ > 0. There are three possibilities (see Figure.) : 5

6 ) V > V +. In this case, the potential decreases, so the particle will cross the potential barrier and gain momentum in order to maintain a constant Hamiltonian. (.4) implies ξ + = (ξ ) + (V V + ). ) V < V + and (ξ ) > V + V. If the kinetic energy of the particle is bigger than the potential jump then the particle will cross the barrier with a reduced momentum. (.4) implies ξ + = (ξ ) (V + V ) 3) V < V + and (ξ ) < V + V. In this case, the kinetic energy is not large enough for the particle to cross the potential barrier, so the particle will be reflected with a negative velocity ξ. [ξ (V + V )] / ξ ) ) ξ [ξ +(V V + )] / ξ 3) ξ Figure. V V + V V + Change of particle momentum across a potential barrier for the case when ξ > 0. If ξ < 0, similar behavior can also be analyzed using the constant Hamiltonian condition (.4). See Fig... The main ingredient in the well-balanced kinetic scheme by Perthame and Simeoni [9] for the shallow water equations with topography was to build in the above mechanism into the numerical scheme in order to preserve the steady state solution of the shallow water equations when the water velocity is zero. Note that the density distribution f remains unchanged across the potential barrier, thus f(t, x +, ξ + ) = f(t, x, ξ ) (.5) at a discontinuous point x of V (x), where ξ + and ξ are related by the constant Hamiltonian condition (.4). This was used in constructing the numerical flux in [9]. 6

7 In this paper, we use this mechanism for the numerical approximation to the Liouville equation with a discontinuous potential. This approximation, by its design, maintains a constant Hamiltonian up to approximation error across the potential barrier. The new issue faced here, not explored in [9], is the discretization in the ξ-direction. Given ξ as a grid point, the ξ + constructed from the constant Hamiltonian condition (.4) may not be a grid point, thus some appropriate interpolations in the ξ-direction is needed here. The approximation in the ξ-direction, and its consequent numerical properties, constitutes the main body of this paper. 3 Scheme I: a finite difference approach 3. The Hamiltonian-preserving numerical flux We now describe our first finite difference scheme for the Liouville equation with a discontinuous potential. We call this scheme as Scheme I. Assume that the discontinuous points of potential V are located at the grid points. Let the left and right limits of V at point x i+/ be V + i+ Note that if V is continuous at x j+/, then V + i+ piecewise linear function V (x) V + i / + V i+/ V + x i / = V i+ and V respectively. i+. We approximate V by a (x x i / ). We will adopt the flux splitting technique used in [9]. The semidiscrete scheme (with time continuous) reads t f ij + ξ j f i+,j f+ i,j x V V + i+ i f i,j+ f i,j x ξ = 0, (3.) where the numerical fluxes f i,j+ are defined using the upwind discretization. Since the characteristics of the Liouville equation may be different on the two sides of a potential discontinuity, the corresponding numerical fluxes should also be different. The essential part of our algorithm is to define the split numerical fluxes f i+,j, f+ i,j at each cell interface. We will use (.5) to define these fluxes. Assume V is discontinuous at x i+/. Consider the case ξ j > 0. Using upwind scheme, f = f i+ ij. However,,j f + i+/,j = f(x+ i+/, ξ+ j ) = f(x i+/, ξ j ) while ξ is obtained from ξ + j = ξ j from (.4). Since ξ may not be a grid point, we have to define it approximately. The first approach is to locate the two cell centers that bound this velocity, then use a linear interpolation to evaluate the needed 7

8 numerical flux at ξ. The case of ξ j < 0 is treated similarly. The detailed algorithm to generate the numerical flux is given below. Algorithm I ξ j > 0 f i+,j = f ij, if V > V +, i+ i+ ( ) if ξ j > V V +, i+ i+ ( ) ξ = ξ j + V + V i+ i+ if ξ k ξ < ξ k+ for some k then f + i+,j = ξ k+ ξ ξ f ik + ξ ξ k f ξ i,k+ else f + i+,j = f i+,k where ξ k = ξ j end if V < V + i+ i+ ( ) ξ = ξ j + V + V i+ i+ if ξ k ξ < ξ k+ for some k if V i+ end ξ j < 0 then f + i+,j = ξ k+ ξ ξ = V + i+ f + i+,j = f i+,j f + i+,j = f i+,j, if V i+ < V +, i+ ( if ξ j > V + i+ ( ξ + = ξ j + f ik + ξ ξ k f ξ i,k+ ) V, i+ V i V + i+ if ξ k ξ + < ξ k+ for some k 8 )

9 then f = ξ k+ ξ + f i+,j ξ i+,k + ξ+ ξ k f ξ i+,k+ else f = f i+,j ik where ξ k = ξ j end if V i+ if V i+ end > V + i+ ( ξ + = ξ j + V V + i+ i+ if ξ k ξ + < ξ k+ for some k then f i+,j = ξ k+ ξ + ξ = V + i+ f i+,j = f+ i+,j ) f i+,k + ξ+ ξ k f ξ i+,k+ The above algorithm for evaluating numerical fluxes is of first order. One can obtain a second order flux by incorporating the slope limiter, such as van Leer or minmod slope limiter [5, 6], into the above algorithm. This can be achieved by replacing f ik by f ik + xs ik, and replacing f i+,k by f i+,k xs i+,k in the above algorithm for all the possible index k, where s ik is the slope limiter in the x-direction. After the spatial discretization is specified, one can use any time discretization for the time derivative. 3. Positivity and l contraction Since the exact solution of the Liouville equation is positive when the initial profile is, it is important that the numerical solution inherits this property. We only consider the scheme using the first order numerical flux, and the forward Euler method in time. Without loss of generality, we consider the case ξ j > 0 < V + i and V i+ conclusion). The scheme reads f n+ ij f ij t for all i (the other cases can be treated similarly with the same + ξ j f ij (c f i,k + c f i,k+ ) x V V + i+ i f ij f i,j x ξ where c, c are positive and c + c =. We omit the possible superscript n of f. The above scheme can be rewritten as V V + f n+ ij = ξ j λ t x i+ i λ t ξ f ij + ξ j λ t x x (c f i,k + c f i,k+ ) V V + i+ + i λ t ξ x f i,j. (3.) 9 = 0,

10 Now we investigate the positivity of scheme (3.). This is to prove that if f n ij 0 for all (i, j), then this is also true for f n+. Clearly one just needs to show that all the coefficients before f n are non-negative. A sufficient condition for this is clearly or V V + ξ j λ t x i+ i λ t ξ x 0, max j ξ j t x + V i+ max V + i i x ξ. (3.3) This CFL condition is similar to the CFL condition (.3) of the usual finite V i+ difference scheme except that the quantity V + i x now represents the gradient of potential at its smooth point, which has a finite upper bound. Thus our new scheme has a hyperbolic CFL condition. According to the study in [8], our second order scheme, which incorporates slope limiter into the first order scheme, is positive under the half CFL condition, namely, the constant on the right hand side of (3.3) is /. The above conclusion are analyzed based on forward Euler time discretization. One can draw the same conclusion for the second order TVD Runge-Kutta time discretization [4]. The l -contracting property of this scheme follows easily with the same hyperbolic CFL condition, because the coefficients in (3.) are positive and the sum of them is. 4 Scheme II: a finite volume approach In this section another flux which results in an l -contracting scheme is proposed. We call this scheme Scheme II. By integrating the Liouville equation (.) over the cell [x i /, x i+/ ] [ξ j /, ξ j+/ ], one gets the following equation t f ij + ξ j f i+,j f+ i,j x V V + i+ i f i,j+ f i,j x ξ = 0. (4.) The upwind discretization depends on the sign of ξ j and V i+ V + i. To illustrate x the basic idea, we assume ξ j > 0, V i+ V + i < 0 and V < V + (this is the case x i+ i+ 0

11 when the particle loses momentum from left to right at the barrier). In this case f i+,j = f i,j+ = ξj+ ( ) ξf x, ξ, t dξ, ξ j ξ i+ ξ j V i+ V + i xi+ x i V x f ( ) x, ξ, t dx j+ By using the condition (.5): f + = ξj+ ( ) ξf x +, ξ, t dξ, = ξj+ ( ) ξf x, ξ, t dξ, (4.) i+,j ξ j ξ i+ ξ ξ j j ξ i+ ξ j where f is defined as f ( ) x, ξ, t = f i+ Using a change of variable on (4.) leads to f + ξj+ ( = ξf x, i+,j ξ j ξ i+ ξ j = ξ j ξ ξ ξ ξf ( ( ) ) x, ξ + V + V, t. i+ i+ i+ ξ + ( ) ) V + V, t dξ i+ i+ ( ) x, ξ, t dξ. (4.3) i+ where ξ = ξ + (V + V ), ξ j i+ i+ = ξ + (V + V ). (4.4) j+ i+ i+ The integral in (4.3) will be approximated by a quadrature rule. Since the end point ξ and ξ may not be grid points in the ξ-direction. We first need to locate the grid points that bound ξ and ξ. There are two possibilities,namely, either ξ and ξ fall into the same cell, or they are in separate cells. In the former case, we use the midpoint rule. In the second case, the composite midpoint rule is used. We propose the following evaluation of the split fluxes f ± in (4.) i+,j Algorithm II if ξ j > 0 f i+,j = f ij, if V i+ > V + if ξ j i+, ( ) V V +, i+ i+

12 ξ = ξ j ξ = ξ j+ ( ) V V + i+ i+ ) ( V V + i+ i+ if ξ k ξ < ξ ξ k+ for some k f + = ξ i+ ξ ξ,j ξ j ξ kf ik else ξ k k, s ξ < ξ k+ < < ξ k+s < ξ ξ k+s+ for some { f + = ξk+ ξ i+,j ξ j ξ ξ k f ik + ξ k+ f i,k+ + } +ξ k+s f i,k+s + ξ ξ k+s ξ ξ k+s f i,k+s end else f + i+,j = f i+,k where ξ k = ξ j end if V i+ < V + i+ ξ = ξ + j ξ = ξ j+ + ( ) V + V i+ i+ ) ( V + V i+ i+ if ξ k ξ < ξ ξ k+ for some k f + = ξ i+ ξ ξ,j ξ j ξ kf ik else ξ k end if V i+ end if ξ j < 0 ξ < ξ k+ < < ξ k+s < ξ ξ k+s+ { f + = ξk+ ξ i+,j ξ j ξ ξ k f ik + ξ k+ f i,k+ + } +ξ k+s f i,k+s + ξ ξ k+s ξ ξ k+s f i,k+s = V + i+ f + i+,j = f i+,j f + i+,j = f i+,j, for some k, s

13 if V i+ < V +, i+ if ξ j+ > ξ = ξ j + ( ) V + V, i+ i+ ( V V + i+ i+ ( ξ = ξ + V V + j+ i+ i+ if ξ k ξ < ξ ξ k+ for some k f = ξ i+ ξ ξ,j ξ j ξ kf i+,k else ξ k k, s ξ < ξ k+ ) ) < < ξ k+s < ξ ξ k+s+ for some { f = ξk+ ξ i+,j ξ j ξ ξ k f i+,k + ξ k+ f i+,k+ + } +ξ k+s f i+,k+s + ξ ξ k+s ξ ξ k+s f i+,k+s end else f i+,j = f ik where ξ k = ξ j end if V i+ > V + i+ ξ = ξ j + ( ) V V + i+ i+ ) ( ξ = ξ + V V + j+ i+ i+ if ξ k else ξ k end if V i+ end ξ < ξ ξ k+ f i+,j = ξ j ξ ξ ξ ξ kf i+,k ξ < ξ k+ for some k < < ξ k+s < ξ ξ k+s+ { f = ξk+ ξ i+,j ξ j ξ ξ k f i+,k + ξ k+ f i+,k+ + } +ξ k+s f i+,k+s + ξ ξ k+s ξ ξ k+s f i+,k+s = V + i+ f i+,j = f+ i+,j for some k, s 3

14 end Remark 4.. The above Algorithm uses a first order quadrature rule at the ends of the interval (4.3), thus it is of first order even if the slope limiters in x-direction are incorporated into the algorithm. One can also use a second order quadrature rule at the ends of intervals (4.3). 5 The schemes in higher dimensions Our D schemes can be easily extended to higher dimension using a dimension-bydimension approach. For example, consider the D Liouville equation f t + ξf x + ηf y V x f ξ V y f η = 0. (5.) We employ a uniform mesh with grid points at x i+, y j+, ξ k+, η l+ in each direction. The cells are centered at (x i, y j, ξ k, η l ) with x i = (x i+ + x i ), y j = (y j+ + y j ξ ), k = (ξ k+ + ξ k η ), l = (η l+ + η l ). The mesh size is denoted by x = x i+ x i, y = y j+ y j, ξ = ξ k+ ξ k, η = η l+ η l. We define the cell average of f as f ijkl = xi+ yj+ ξk+ ηl+ f(x, y, ξ, η, t)dηdξdydx. x y ξ η x i y j ξ k η l Similarly to the D case, we approximate the potential by a piecewise bilinear function, and for convenience, we always provide two interface values of potential at each cell interface. When the potential is smooth at a cell interface, the two potential interface values are identical. The D Liouville equation (5.) can be semi-discretized as t f ijkl f i+ + ξ,jkl f+ f i,jkl i,j+ k + η,kl f+ i,j,kl l x y V V + i+,j i,j f ij,k+,l f ij,k,l x ξ V V + i,j+ i,j f ijk,l+ y f ijk,l η = 0, where the interface values f ij,k+,l, f ijk,l+ are provided by the upwind approximation, and the split interface values f i+,jkl, f+ i,jkl, f i,j+,kl, f+ can be obtained i,j,kl using essentially the same algorithm described in subsection.3 or.5 for the D case. Since the gradient of the potential at its smooth points are bounded, this scheme similar to the D scheme, is also subject to a hyperbolic CFL condition under which the scheme is positive, and Hamiltonian preserving (if the discontinuity of V aligns with the grids). 4

15 6 The l -stability theory of Scheme I In this section we prove the l -stability of Scheme I (with the first order numerical flux and the forward Euler method in time) under a suitable assumption on the initial data. We also show that if this assumption does not hold, the numerical solution can grow unbounded in l. We consider the simple case when V (x) is a step function, with a jump D, D > 0 at x m+. Namely V m+ V + m+ = D, V ± i+ = V, i < m, V ± m+ i+ = V +, i > m. m+ We consider the typical situation that ξ < D, ξ M > D, so that all the three situations discussed in Section are included. We also choose the mesh such that 0 is a grid point in the ξ-direction. Define an index set Dl 4 = {(i, j) x i x m, ξ j < ξ D}. Due to velocity change across the potential jump at x m+, Dl 4 represents the area where particles come from outside of the domain [x, x N ] [ξ, ξ M ]. In order to implement Scheme I conveniently, we need to choose the computational domain as E d = {(i, j) i =,, N, j =,, M} \ D 4 l. (6.) Figure 6. depicts E d and D 4 l. We define the l -norm of a numerical solution f ij to be f = f ij N d (i,j) E d with N d being the number of elements in E d. 6. l -stability under an assumption on the initial data Given the initial data f 0 ij, (i, j) E d. Denote the numerical solution at time T to be f L ij, (i, j) E d. To prove the l -stability, we need to show that f L C f 0. Due to the linearity of the scheme, the equation for the error between the analytical and the numerical solution is the same as (3.), so in this section, f ij will denote the error. We assume there is no error at the boundary, thus f n ij = 0 at the boundary. If the l -norm of the error introduced at each time step in incoming boundary cells is ensured to be o() part of f n, our following analysis still applies. Since V x (x) = 0 except at x = x m+/, Scheme I is given by: ) if ξ j > 0, i m +, f n+ ij = ( ξ j λ t x )f ij + ξ j λ t x f i,j ; (6.) 5

16 ) if ξ j < 0, i m, f n+ ij = ( ξ j λ t x )f ij + ξ j λ t x f i+,j ; (6.3) 3) if ξ j > D, 4) if 0 < ξ j D, 5) if ξ j < 0, f n+ m+,j = ( ξ jλ t x )f m+,j + ξ j λ t x (c j,kf m,k + c j,k+ f m,k+ ) ; (6.4) f n+ m+,j = ( ξ jλ t x)f m+,j + ξ j λ t xf m+,k ; (6.5) f n+ mj = ( ξ j λ t x)f mj + ξ j λ t x(c jk f m+,k + c j,k+ f m+,k+ ), (6.6) where 0 c jk and c jk +c j,k+ =. In (6.4) k is determined by ξ k ξj D < ξ k+, in (6.6) ξ k = ξ j, and in (6.5) ξ k ξj + D < ξ k+. We omit the superscript n of f ij on the right hand side. Using the triangle inequality in (6.)-(6.6), one typically gets the following f n+ α ij fij n, (6.7) N d (i,j) E d where the coefficients α ij are positive. One can check that, under the hyperbolic CFL condition (3.3), α ij except for possibly (i, j) D m D 4 m+ with the definitions: D m = {(m, j) 0 < ξ j < ξn D + ξ}, D 4 m+ = {(m +, j) ξ j < ξ /4 + D + ξ}. 6

17 v M D m E d D l 4 4 D m+ v x x m+/ Figure 6. Sketch of the index sets D m, D4 m+, D4 l. x N Denote M = max α ij, M = max α ij. (i,j) Dm (i,j) Dm+ 4 Our next step is to prove that M, M are bounded independent of the mesh size. Let us first examine M. Define the set { Sj m = j ξj > ξ } D, j D ξ j < ξ for (m, j) Dm. Let the number of elements in Sj m be Nj m. One can check that Nj m because every two elements j, j Sm j satisfy ξ j D ξ j D ξ j ξ j ξ. On the other hand, one can easily check from (6.4), for (m, j) D m, α mj < + N m j 3, so the boundedness of M is proved. Next we study M. Define the set S m+ j = {j ξ j + D ξ j < ξ } for (m +, j) D 4 m+. 7

18 Let ξ max = max{ ξ, ξ M }. Using the CFL condition, ξmax t x. So from (6.6) one can get, for (m +, j) D 4 m+, the estimate for α m+,j : α m+,j < + j S m+ j ξ j λ t x + ξ max j S m+ j ξ j. (6.8) Since ξ j, for j S m+ j, are in fact an arithmetic progression with increment ξ. Denote the minimum and maximum element in S m+ j to be m, m respectively. Since ξ m, ξ m < 0, one has ξ m ξ m. The last summation in (6.8) turns out to be j S m+ j ξ j = ξ m + ξ m ( ξm ξ m ξ ) + ξ m ξm ξ On the other hand, because m, m Sj m+, ξm + D ξm + D ξ, ξm + D ξm + D + ξ, ξm ξm + ξm + D ξ + 4 ξ, ξ m ξ m ξ Combine (6.8)-(6.0), we get + ξ max. (6.9) ξ m + D + ξ ξ max + ξ. (6.0) α m+,j < + ξ max (ξ max + ξ) = 3 + ξ ξ max with (m +, j) D 4 m+. Therefore the boundedness of M is proved. In summary, we have M < 3, M < 3 + ξ ξ max, both are bounded independent of mesh size. Denote M = max(0, M ), M = max(0, M ). From (6.7), f n+ f n + M N d We now impose an assumption: Assumption (i,j) D m f n ij + M N d (i,j) D 4 m+ f n ij. (6.) 8

19 There exists a positive constant ξ z such that it holds that (i, j) S z = {(i, j) x i < x m+, 0 < ξ j < ξ z }, (6.) f 0 ij C f 0. (6.3) Remark: The semiclassical limit initial data (.4) does not satisfy this condition. Thus Scheme I, when directly applied to this problem, may have stability problems, as shown in the next subsection. However, if the decomposition idea mentioned in the Introduction is used, Scheme I is still suitable, which is what will be done in the numerical experiments of Section 9. We now establish the following theorem: Theorem 6.. Under Assumption, there exists an h 0 > 0, when x < h 0, the l -stability property of the scheme (6.)-(6.6) holds. Proof. From (6.), f L f 0 + M N d It remains to estimate and L n=0 We begin with estimating S. Define the set (i,j) D m f L C f 0 L S = n=0 L S = n=0 fij n + M L (i,j) D m (i,j) D 4 m+ N d n=0 fij n f n ij (i,j) D 4 m+ S r = {(i, j) x i > x m+, (m +, j) Dm+ 4 }. fij n. (6.4) (6.5). (6.6) (i, j) S r, due to the zero boundary condition and the upwind nature of the scheme, one has f n ij = (p,q) S r β ijn0 pq f 0 pq, (i, j) S r (6.7) 9

20 with βpq ijn0 0. Notice Dm+ 4 S r, S L (p,q) S r where we have defined n=0 (i,j) Dm+ 4 βpq ijn0 fpq 0 F(p, q) fpq 0, (6.8) (p,q) S r F(p, q) = L n=0 (i,j) Dm+ 4 β ijn0 pq, (p, q) S r. (6.9) The next step is to estimate these coefficients. Define then (6.9) gives β ij0 F(p, q) = pq = Hence it is useful to evaluate β ij0 n=0 (i,j) Dm+ 4 n=0 pq. β ijn0 pq, (p, q) S r, L β ijn0 pq (i,j) D 4 m+ β ij0 pq. Notice βpq ij0 is not zero only when p i and q = j due to the upwind flux and constant potential. We first evaluate βpq ij0 when p = i and q = j. Denote c j = ξ j t, x cj = ξ j t. From scheme (6.3) x β ij0 ij = n=0 β ijn0 ij = Since (i, j) S r and c j Dλ t x, so n=0 (c j ) n = c j = c j. (6.0) c j Dλ t x λ. We now evaluate β ij0 pj when p > i. From scheme (6.3), β ij,n+,0 pq = c j β ijn0 pq + c j β i+,jn0 pq, (6.) then a sum of n from 0 to in (6.) gives β ij0 pq = β i+,j0 pq, i < p. (6.) 0

21 We now can evaluate F(p, q) for (p, q) S r. F(p, q) βpq ij0 = βm+,q,0 pq = βpq m+,q,0 = = βpq p,q,0 λ. (6.3) (i,j) D 4 m+ Therefore, from (6.8) we get S F(p, q) fpq 0 λ (p,q) S r λ Our next step is to estimate S. Define the set Similarly when (i, j) S l, one has (p,q) S r f 0 pq (p,q) E d f 0 pq = λ N d f 0. (6.4) S l = {(i, j) x i < x m+, (m, j) Dm }. f n ij = (p,q) S l γ ijn0 pq f 0 pq, (i, j) S l. (6.5) Dividing set D m into two parts: D, m = {(i, j) D m ξ j ξ z }, D, m = {(i, j) D m ξ j < ξ z }, and also define the corresponding two parts of S l S l = {(i, j) S l ξ j ξ z }, S l = {(i, j) S l ξ j < ξ z }. Note that S l is a subset of S z in (6.). Correspondingly, S is also divided into two parts L L S = fij n + fij n = S + S. (6.6) n=0 (i,j) D, m n=0 (i,j) D, m Similar to the previous case, we can get the upper bound of the first term with λ ξ zλ. Substituting (6.5) into S t gives x S L (p,q) S l S λ N d f 0 (6.7) n=0 (i,j) D, m γpq ijn0 f 0 pq.

22 Using Assumption, Now we evaluate (p,q) S l S C f 0 (p,q) S l L = C f 0 γ ijn0 f n ij = n=0 L (i,j) D, m n=0 (i,j) Dm, (p,q) S l γpq ijn0 γpq ijn0 pq when (i, j) Dm,. Write (6.5) as (p,q) S l. (6.8) γpq ijn0 fpq 0, (i, j) D, m. (6.9) When the initial values are constant, including the ghost cells at the boundary, the numerical solutions at the next time step still remain unchanged, while the coefficients γ ijn0 pq Continuing from (6.8), with λ 3 = C T (x N+ x )λ t x in (6.9) do not include those corresponding to the ghost cells, thus (p,q) S l L S C f 0 = γ ijn0 pq, (i, j) D, m. n=0 (i,j) D, m C f 0 LN d N C TN d f 0 (x N+ x )λ t λ 3 N d f 0 (6.30) x being an O() quantity. Now from (6.6), (6.7) and (6.30), Combing (6.4), (6.4) and (6.3), S (λ + λ 3 )N d f 0. (6.3) f L f 0 + M λ f 0 + M (λ + λ 3 ) f 0 = [ + M λ + M (λ + λ 3 )] f 0 C f 0 where C + M λ + M (λ + λ 3 ). Thus Theorem is proved.

23 6. An unstable example There arises another question about whether condition (6.3) in Assumption is necessary for the l -stability. In this subsection we give an counter example which shows that if this condition is violated, the solution may become unbounded in l norm. Here we impose the assumption: Assumption There exists a positive constant ξ z such that (i, j) S z in (6.), it holds that f 0 ij C f 0 x q, q > 0 (6.3) with C independent of the mesh size. Remark: Assumption reduces to Assumption in the case q = 0. We first introduce some notations. Define the sets S m = {k D + ξ ξ k 3 0D ξ}, S m = {k j S m, s.t. ξ k ξ j < ξ or ξ k = ξ j + ξ}. Let N s be the number of elements in S m. We name the elements in S m as k i, i =,,, N s such that k < k <... < k Ns. Define an one-to-one map from S m to S m as T s (k) = j s.t. j S m, ξ k ξ j ξ, k S m. It is clear that ξ Ts(k i ) D + i ξ, i =,,, N s. Let q = min( q, ). We choose T such that Tλx t < x m+, thus L < m and L < N m. Let x N+ x m+ x and Tλ x t < where int(x) is the biggest integer equal to or less than x. Define a function G(k) as L 0 = int(l q )L, (6.33) G(k) = int(lξ k λ t x ), k S m. Clearly, G(k ) G(k ) G(k Ns ). Since L < m, so G(k) < m for k S m. Define the following set of index of cells H = {(i, j) j S m, m G(j) < i m}. Let N h be the number of elements in H. Our next step is to check the condition under which N h > L 0. 3

24 Lemma 6.. N h > L 0 under the mesh size restrictions D x < 96, (6.34) 0λ ξ x x < 4 0( 0 3) Tλ t x Dλ ξ x Tλ t x +, (6.35) λt ξ λ t x λt ξ x < λx( t 0T ) q. (6.36) q Proof. According to the definitions, N h = G(k) = int ( ( ) ) Lξ k λ t x > Lλ t x ξ k N s k S m k S m k S m ) > Lλ t x (ξ k D ξ ( 0 3) D ξ 3 > Lλ t x > = > > k S m N s i= 3L 0D λ t x ( ( ) D + i ξ) D 3 ( ) ( L λ t x + λt ) ξ 0 3 D T [ N s ( ) ] ( ) D + i ξ D + i ξ D i= ( ) ( 3 L λ t x + λt ) ξ 0 3 D T [ Lλ t N s ( ) ] ( ) x 3 D + i ξ D + i ξ D ξ 0D ξ i= ( ) ( 3 L λ t x + λt ) ξ 0 3 D T Lλ t 0D 4 ξ 3 x 3x x 0D ξ Ddx ( D 3 L λ t x + λt ξ Lλ t x 0D ξ 3 D 8 0D ξ 3 T 3 L (λ t x + λt ξ T We impose the following restriction on the mesh sizes ) ( 0 3 ) D ) ( 0 3 ) D (6.37) 8 0D ξ 3 < ( + λt ξ ) ( 0 3) D ξ < Tλ t x 3 D 6, (6.38) 0, (6.39) 4

25 then continue from (6.37), N h > Lλ t x 0 ξ = Lλt x λt ξ 0 t. According to (6.33), L 0 < LL q = LT q. Therefore, in order that N ( t) q h > L 0, one needs to impose the mesh size restriction ( λ t ) x t < λt ξ q. (6.40) 0T q One can rewrite the mesh size restriction (6.38), (6.39) and (6.40) to that on x which are (6.34)-(6.36). Now, under the mesh size restriction (6.34)-(6.36), it holds that N h > L 0. We now prove the following theorem: Theorem 6.. q > 0 in Assumption, h 0 > 0, x < h 0, T > 0, B > 0, fij 0, (i, j) E d satisfying Assumption, such that f L > B f 0. Proof. We define a function F H in H as s F H (i, j) = m G(j) + + G(k l ) if j = k s, (i, j) H. l= F H in fact is an one-to-one map from H to (,,, N h ). Now define the set H L = {(i, j) (i, j) H, F H (i, j) L 0 }. Since N h > L 0 by Lemma 5., the number of elements in H L is L 0. We can now introduce the initial value fij 0 satisfying the condition of Theorem : fij 0 = c 0, (i, j) H L, (6.4) fij 0 = 0, (i, j) E d \ H L, (6.4) where c 0 > 0 is a constant. 5

26 We first check that these initial values satisfy Assumption. Since f 0 ij f 0 = N d L 0 < MN thus Assumption is satisfied if (x N+ = (x N+ L q q x)(ξ M+ ξ )λ t x, λ ξ xt q x q q x)(ξ M+ ξ )λ t x < C λ ξ xt q x q x q. (6.43) Condition (6.43) is satisfied under the following mesh size restriction x < ( (x N+ because we have chosen q < q. C λ ξ xt q x)(ξ M+ ξ )λ t q x ) q q (6.44) Next we analyze the relation between f L and f 0. Since L < N m, the solution at the boundary cells remains zero for all the time steps. If we define the sets S m m = {(i, j) i = m, j S m}, Sm l = {(i, j) x i < x m+, j S m }, Sm r = {(i, j) x i > x m+, j S m }, then at each time step, only solutions at cells belonging to S l m or Sr m nonzero. Namely are possibly fij n = 0 for (i, j) E d \ {Sm l Sr m }. (6.45) Since our scheme is positive preserving, and the initial values (6.4) and (6.4) are nonnegative, the numerical solutions at each time step are always nonnegative. Then similar to the proof of (6.7), at each time step f n+ = N d = N d = N d f n+ ij (i,j) E d (i,j) E d α ij f n ij α ij f n ij + N d α ij f n ij + N d (i,j) S l m (i,j) S r m (i,j) E d \{S l m Sr m } f n ij, (6.46) 6

27 Note the last term in (6.46) is zero by (6.45). For scheme (6.) one sees that among those α ij with (i, j) S l m S r m, α ij only when (i, j) Sm m, so continuing from (6.46) gives f n+ = N d α mj f n mj + N d f n ij. (6.47) (m,j) S m m (i,j) E d \S m m We now estimate α mj for (m, j) Sm m. From schemes (6.) and (6.4), for (m, j) S m m, by setting j = T s (j), one has where c j j are the coefficients in (6.4). α mj = ξ j λ t x + ξ j λt x c j j, (6.48) According to the definitions of S m and S m, c j j, ξ j < 3 D, ξj > D in (6.48). So (6.48) gives α mj > + Then (6.49) together with (6.47) give D 6 λt x. (6.49) f n+ > = D 6 λt x N d (m,j) Sm D 6 λt x N d (m,j) Sm f n mj + N d (i,j) E d f n ij f n mj + fn. (6.50) Summing up (6.50) from n = 0 to L, one gets Write f L > f 0 + f n ij = (p,q) S l m D L 6 λt x N d n=0 (m,j) Sm Since Sm m Sm, l substituting (6.5) into (6.5) gives f L > f 0 + Dλ t L 6 x N d (p,q) Sm l n=0 = f 0 + Dλ t L 6 x N d f n mj. (6.5) η ijn0 pq f 0 pq, (i, j) Sl m. (6.5) (p,q) H L 7 (m,j) S m m n=0 (m,j) Sm ηpq mjn0 ηpq mjn0 f 0 pq c 0

28 = f 0 + Dλ t c 0 6 x N d (m,j) Sm f 0 + Dλ t c 0 s 6 x L N d l= L η mjn0 pj (p,j) H L n=0 (p,k l ) H n=0 η mk ln0 pk l, (6.53) where k s S m is the quantity such that i satisfying m G(k s ) < i m and F H (i, k s ) = L 0. Thus we need to estimate L (p,j) H n=0 ηmjn0 pj for (m, j) Sm m. From scheme (6.), one has for (k, j) H, η kj,n+,0 pj Adding (6.54) from n = 0 to L leads to therefore, L η kjl0 pj + n=0 n=0 η kjn0 pj L η kjn0 pj = = ( ) ξ j λ t x η kjn0 pj + ξ j λ t xη k,jn0 pj. (6.54) = ( ξ j λ t x n=0 ) L n=0 η kjn0 pj + ξ j λ t x L η k,jn0 pj η kjl0 ξ j λ t pj x L n=0 η k,jn0 pj, L = η k,jn0 pj [η kjl0 ξ n=0 j λ t pj + η k,jl0 pj ] x = = = L n=0 n=0 η pjn0 pj ξ j λ t x L η kjn0 kj ξ j λ t x k l=p+ k l=p Applying (6.55) when k = m, one gets the relation L n=0 L η mjn0 pj = n=0 η ljl0 pj η mjn0 mj m ξ j λ t x For a fixed j S m, adding (6.56) for p such that (p, j) H gives L (p,j) H n=0 η mjn0 pj L = G(j) n=0 η mjn0 mj ξ j λ t x m η kjl0 lj. (6.55) l=p l=m G(j)+ η mjl0 lj. (6.56) (l m + G(j))η mjl0 lj. (6.57) 8

29 According to the definition of S m and S m, when j S m, ξ j < 3 D. The CFL condition (3.3) implies that Dλ t x <, so ξ j λ t x < 3 when j S m. Define µ j = ξ j λ t x, one has ηmjl0 lj = ( µ j ) L+l m µ m l j C m l L, hence, m l=m G(j)+ η mjl0 lj = = m l=m G(j)+ G(j) l= ( µ j ) L+l m µ m l j ( µ j ) L l µ l jc l L = C m l L int(µ j L) l= ( µ j ) L l µ l jc l L <. The proof of the last inequality is in the Appendix A. Continuing from (6.57) gives L (p,j) H n=0 η mjn0 pj L > G(j) η mjn0 mj G(j) ξ j λ t x n=0 = G(j) ( ξ jλ t x) L ξ j λ t x G(j) ξ j λ t x. (6.58) By the definitions of S m and S m, for j S m, ξ j > ( D + ξ) D ξ > D ξ ξ. (6.59) In order for ξ j λ t x > = λt ξ ξ, from (6.59), L T D ξ ξ > ξ < D + ) x < ( Tλ ξ x Tλ ξ x Under mesh size restriction (6.60), ξ j λ t x > L > L, thus By using (6.58), one gets ξ D. (6.60) ( + ) λ ξ Tλ ξ x x ( ξ j λ t x )L < ( L )L < e <.5. L (p,j) H n=0 η mjn0 pj > G(j) 0ξ j λ t x > L ξ j λ t x 0 > L 0, (6.6) 9

30 where in the last inequality we used ξ j λ t x > under mesh size restriction (6.60). L Next one needs to estimate s appeared in (6.53) as the upper bound of the summation. From the definition of s in (6.53), s G(k l ) L 0. (6.6) l= On the other hand, for s N s, s l= G(k l ) < Lλ t x s < Lλt x 3 D < < l= s Lλ t x 3 D ξ Lλ t x 3 D ξ ξ kl + s < Lλ t x s l= 3ξ k l ξk D + s λ t x T l l= D+(s +) ξ D ξ k D + s λ t xl ξ + s l λ t ξ + s 3ξ ξ Ddξ + s λ t xt λ t ξ [ ]3 D (s + ) ξ + s λ t x T λ t ξ + s + s. (6.63) By choosing s = s N 5 6 q, (6.63) gives s l= G(k l ) < 8λt x(d) 4L ξ s 3 + s λ t xt 3 λ t ξ < 8λt x 4L ξ (D) 3 < 8λt x(d) 4 TL 3 λ t ξ L ( 0D ) λ t x T + = + + s s N q + s Nλ t x T ( 0D 3 D ξ ( 0D λ t ξ + s N )3 5 4 q D 3 ξ ) D 3 + λ t ξ ξ ( 8λ t λ t ( 0 ) x (D) 4 T ξ D )3 5 4 q T 3 L 5 4 q 3 λ t ξ ( 0D ) ( 0D ) D λ t D λ t 3 ξ L 3 xl +. (6.64) T 30

31 If one imposes the mesh size restrictions which corresponds to ( 8λ t x(d) λ t ( 0 4 T ξ T 3 λ t ξ ( 0D ) D 3 ( 0D ) D λ t 3 ξ L T x < T λ t x 3 λ t x L < 4 L q, < 4 L q, ) D ) q 3 λ t ξ ( λ 3λ t t ( 0 x (D) 4 T ξ T x < T ( λ t 0D x 4 ) D 3 t < T T ( λ t 0D x 4 ) D 3 then under (6.65)-(6.67) one has, for s = s N 5 6 q, s l= Comparing (6.68) with (6.6) gives s > s 5 N 6 q > λ t x λ t ξ L 5 4 q < 4 L q, ) D ) q 4 q, (6.65) q, (6.66) q, (6.67) G(k l ) < 3 4 L q < L 0. (6.68) ( 0D ) D 3 T λ t ξ 5 6 q L 5 6 q +. (6.69) Now combining with (6.6) and (6.69), (6.53) gives ( 0D D ) D f L > f T f 0 + c 0 0 λt x N d D L 0 c 0 0 λt x N d ( 0D λ t ξ ) D 3 T λ t ξ 5 6 q L 5 6 q 5 6 q L 6 q (6.70) 3

32 = + D 0 λt x ( 0D ) D 3 T λ t ξ 5 6 q L 6 q f0. (6.7) So B > 0, one can choose mesh size such that + D 0 λt x ( 0D ) D 3 T λ t ξ 5 6 q L 6 q > B or x < T λ t ξ D 0B λt x ( 0D ) D 3 T λ t ξ q q, (6.7) under which it holds that f L > B f 0. 7 Stability theory of Scheme II In this section we study the l and l stability of Scheme II. Its positivity is obvious under the hyperbolic CFL condition (3.3). Theorem 7.. If the forward Euler time discretization is used, then the flux is given by Algorithm II yields the scheme (4.) which is l -contracting and l -stable. Proof. For simplicity, we discuss the case when the potential has only one discontinuity at grid point x m+ with jump V V + = D > 0, and V (x) < 0 at smooth m+ m+ points. The other cases, namely, when V (x) 0, or the potential having several discontinuity points with increased or decreased potential jumps, can be discussed similarly. We consider the typical situation when ξ < D, ξ M > D, so that all the three situations discussed in Section are included. We assume the mesh is defined such that 0, ± D are grid points in the ξ-direction. We define some sets of indexes D + m = {(m, j) ξ j > 0}, D + m+ = {(m +, j) ξ j } D, 3

33 Dm {(m, = j) ξ D ξ j { Dm+ = (m +, j) ξ j+ D { Dl 4 = (i, j) x i x m, ξ j+ These domains are shown in Figure (7.). } 0, }, } ξ D. v M + D m+ D m + E d D m D l 4 D m+ v x x m+/ Figure 7. Sketch of the index sets D + m, D+ m+, D m, D m+, D 4 l. x N Again the computational domain is chosen as (6.): E d = {(i, j) i =,, N, j =,, M} \ Dl 4. Now denote F i = x V V +. Our scheme (4.) with Algorithm II can be i+ i made precisely as ) if ξ j > 0, i m +, ) if ξ j < 0, i m, f n+ ij = ( F i λ t ξ ξ j λ t x) fij + F i λ t ξf i,j + ξ j λ t xf i,j ; (7.) f n+ ij = ( F i λ t ξ ξ j λ t x) fij + F i λ t ξ f i,j + ξ j λ t x f i+,j ; (7.) 33

34 3) if ξ j > 0, f n+ m+,j = ( F m+ λ t ξ ξ jλx) t fm+,j + F m+ λ t ξ f m+,j + ξ j λ t x f+ ; (7.3) m+,j 4) if ξ j < 0, f n+ mj = ( F m λ t ξ ξ j λx) t fmj + F m λ t ξ f m,j + ξ j λ t x f, (7.4) m+,j where we omit the superscript n on the right hand side. By summing up (7.)-(7.4) for (i, j) E d, one typically gets the following expression (i,j) E d f n+ ij = (i,j) E d α ij f ij + (m+,j) D + m+ ξ j λ t x f+ m+,j + (m,j) D m ξ j λ t x f m+,j I + I + I 3 (7.5) As in the proof of stability of Scheme I, we assume that f satisfies the zero boundary condition. In this situation, the coefficients α ij in (7.5) satisfy α ij, (i, j) E d \ {D + m D m+}, (7.6) α ij ξ j λ t x, (i, j) D+ m D m+. (7.7) We now study the relation between I and (m,j) D + m ξ jλ t x f mj. Let p N+ = ξ N+ D, and assume Assume ξ J = 0 for some J. Since ξ k < p N+ ξ k+ ξ N+. I k ξ j f mj + p N+ ξ k ξ j=j ξ k f mk (m,j) D + m ξ j f mj, thus I ξj λ t x f mj. (7.8) (m,j) D + m Similarly, one gets I 3 ξj λ t x f m+,j. (7.9) (m+,j) D m+ 34

35 Combining (7.5), (7.6), (7.7), (7.8) and (7.9) gives ij. (7.0) (i,j) E d f n+ (i,j) E d f n ij This is the l -contracting property of this scheme. Next we prove the l -stability. In the cells where ξ j > 0, i m+ or ξ j < 0, i m, one uses (7.) or (7.). Observing that the coefficients on the right hand side of (7.) or (7.) are positive and the sum of them is, so in fact these schemes are l - contracting. It remains to study the cells where ξ j > 0, i = m + or ξ j < 0, i = m, corresponding to (7.3) or (7.4). We first consider (7.3). It can be checked that when ξ j < D, f + = m+,j f m+,k with k such that ξ k = ξ j, thus the l -contracting property still holds. When ξ j ξ k D, denote ( ) ( ) ξ = ξ j D, ξ = ξ j+ D. (7.) Since V + m+ ξ < ξ ξ k+ with s. In this case ξ k+s+ < V, one has ξ m+ ξ for any k. Assume ξ k > ξ. Therefore, it is impossible that ξ < ξ k+ { f + = ξk+ ξ ξ m+,j k f mk + ξ k+ f m,k+ + ξ j ξ Substituting (7.) into (7.3) yields f n+ +ξ k+s f m,k+s + ξ ξ k+s ξ < < ξ k+s ξ k+s f m,k+s } m+,j = ( F m+ λ t ξ ξ jλ t x) fm+,j + F m+ λ t ξ f m+,j + λ t x { ξk+ ξ ξ k f mk + ξ k+ f m,k+ + ξ +ξ k+s f m,k+s + ξ ξ k+s ξ ξ k+s f m,k+s } < ξ. (7.). (7.3) Observing that the coefficients on the right hand side of (7.3) is still positive. Thus it remains to check the sum of these coefficients given by I 4 = ξ j λ t x + λ t x { ξk+ ξ ξ k + ξ k+ + + ξ k+s + ξ ξ } k+s ξ k+s ξ ξ 35

36 then Let Also notice = + ξ j λ t x ξ k+ ξ ξ ξ k (ξ + ξ ) = ξ k + ξ k+ + + ξ k+s + ξ ξ k+s ξ ξ k+s ξ j. (7.4) ξ k = ξ k + ξ k+s = ξ k+ + ξ k+s, a = ξ k+ ξ (0, ], ξ a = ξ ξ k+s ξ (0, ], ξ k = ξ k s ξ, ξ k+s = ξ k + s ξ. [( ) ( )] ξ k+ ξ + ξ k+s ξ = (a a ) ξ, hence one gets { ξk+ ξ ξ k + ξ k+ + + ξ k+s + ξ ξ } k+s ξ k+s ξ j ξ ξ { = ξk+ ξ ξ k ξ j ξ + ξ k + + ξ k + ξ ξ } k+s ξ k + s ξ (a ξ ξ a ) j as = (ξ ξ )ξ k ξ j ξ = (ξ ξ )(ξ + ξ ) ξ j ξ = ( ξ j+ + s ξ ξ j (a a ) (ξ ξ j < + ξ 8ξ j < + ξ )( ξ j+ + s ξ ξ j (a a ) (ξ ξ ) ξ j (a a ) ) + ξ j ) + ( a a ) ξ ξ j (a a ) ξ 8 D. (7.5) Substituting (7.5) into (7.4), the sum of the coefficients in (7.3) is estimated I 4 < + ξ j λ t ξ x 8 D < + t, (7.6) d where the last inequality is deduced by using the CFL condition. 8λ t ξ 36

37 Now we consider case (7.4). Denote ( ) ( ) ξ = ξ j + D, ξ = ξ j+ + D. (7.7) In this case, we know ξ ξ < ξ. So there are two cases ξ k or ξ k ξ < ξ k+ < ξ ξ k+ 3 corresponding respectively to ξ < ξ ξ k+ or f = ξ ξ ξ m+,j k f m+,k (7.8) ξ j ξ { f = ξk+ ξ ξ m+,j k f m+,k + ξ ξ k+ ξ j ξ ξ ξ k+ f m+,k+ }. (7.9) Substituting (7.8) or (7.9) into (7.4) gives or f n+ mj = ( F m λ t ξ ξ j λx) t fmj + F m λ t ξ f m,j + λ t ξ ξ x ξ k f m+,k, (7.0) ξ f n+ mj = ( F m λ t ξ ξ j λ t x) fmj + F m λ t ξ f m,j + λ t x { ξk+ ξ ξ k f m+,k + ξ ξ } k+ ξ k+ f m+,k+. (7.) ξ ξ In each case, the coefficients on the right hand side are positive. Thus it remains to check the sum of the coefficients, which is, respectively, { (ξ + ξ j λ t ξ ) ξ } k x ξ ξ j or + ξ j λ t x Let D k be (ξ ξ ) ξ k ξ ξ j or (ξ k+ (7.) { (ξk+ ξ ) ξ k + (ξ ξ } k+ ) ξ k+. (7.3) ξ ξ j ξ ) ξ k +(ξ ξ k+ ) ξ k+ ξ ξ j. One has D k < (ξ ξ )[ ( ξ + ξ ) + ξ] = ξ ξ j ( ξ + ξ ) + ξ ( ξ + ξ ) < + t, λ t ξ D thus the sums (7.) and (7.3) are both bounded above by + λ t ξ t. (7.4) D 37

38 Combining (7.6) and (7.4), and letting C 0 = λ t ξ D be an O() quantity, we get f n+ < ( + C 0 t) f n, thus f L < ( + C 0 t) L f 0 < e C 0T f 0. (7.5) This is the l -stability property of this scheme. Similar to Scheme I, Scheme II is also positive under the CFL condition such as (3.3). 8 Discontinuous solutions and numerical accuracy When the solution of the Liouville equation is smooth, the formally second order shock capturing finite difference scheme will produce second order numerical approximations. Consequently the physical observables obtained by evaluating the numerical δ-integral concentrated on these numerical solution, such as those in (.8)-(.9), should generally be of first order. However, when the potential V is discontinuous, the solution of the Liouville equation, even with smooth initial data, may produce discontinuities at downstream part of the potential discontinuity. These discontinuities influence the accuracy of the numerical δ-integral through which the desired physical observables are obtained. 8. Discontinuities produced in the downstream part For the Liouville equation with a discontinuous potential, if the initial data are smooth, the level set function exhibits discontinuities in the downstream side of the potential discontinuity. We use a D example to illustrate this. Let φ be the level set function that solves the d Liouville equation with the potential given by A, x < b ; V (x) = Ax b, b < x < 0 ; 0, x > 0, (8.) with A, b positive. Let the initial velocity profile be a constant velocity ξ 0 > 0 and the initial density is denoted by ρ 0 (x). The initial value of the level set function is φ(x, ξ, 0) = ξ ξ 0. 38

39 We consider the solution at t = T. In this example the potential is continuous, and the initial level set function is continuous, so this level set function should still be continuous at t = T. Now look at the set S = {(x, ξ 0 ) b < x < 0}, which is part of the initial zero level set. The bicharacteristic of the Liouville equation (.) is { dx = ξ, x(0) = x dt 0 ; dξ = V (8.) (x), ξ(0) = ξ dt 0. Denote the solution of (8.) by then define the set x = x(x 0, ξ 0, t), ξ = ξ(x 0, ξ 0, t), S = {(x, ξ) x = x(x 0, ξ 0, T), ξ = ξ(x 0, ξ 0, T), (x 0, ξ 0 ) S }, which is a subset of the zero level set of φ(x, ξ, T). Select an element (x, ξ ) S. We now want to evaluate φ ξ (x, ξ, T). Assume (x, ξ ) = (x( c, ξ 0, T), ξ( c, ξ 0, T)). Consider the case that T is large enough so that (x, ξ ) is a downstream point. Then the relation between x and c is x (c) = T ξ 0 + Ac b b A ( ξ 0 + Ac b The density at x = x at time T is the inverse of the Jacobian of x(c) multiplied by the initial density ρ(x, T) = ρ 0( c) bρ 0 ( c) dx (c) =. dc b On the other hand, it is known So we have ρ(x, T) = ρ 0 ( c) ξ +ɛ ξ ɛ TA ξ 0 +Ac b δ(φ(x, ξ, T))dξ = φ ξ (x, ξ, T) = ta ξ0 + Ac b ). ρ 0 ( c) φ ξ (x, ξ, T). b b. If we take the limit b 0, we know φ ξ (x, ξ, T). Moreover, the time needed for (x, ξ ) to be in the downstream domain, which is the time for the point ( c, ξ 0 ) to reach x = 0 along its bi-characteristic, should be T(c) = b ξ A 0 + Ac. Notice T(c) 0 as b 0, we know in this example when b taking limit to the discontinuous potential, the level set function should only contain discontinuities in the downstream domain. 39

40 8. Influence of discontinuities on the accuracy of the numerical evaluation of moments In the previous subsection, we showed that the discontinuities inevitably emerge in the downstream part of the potential discontinuity. As is well known, the l - convergence rate for finite difference schemes to compute a discontinuous solution of a linear equation is at most halfth order [4], [7]. Here we show that a halfth order error is also introduced when evaluating the moments (.8) (.9) based on the discontinuous part of the solution. We use the D linear advection equation with the Riemann initial data to illustrate this. Consider equation u t + au x = 0 (8.3) with initial data u(x, 0) = {, x < 0 ;, x > 0. (8.4) Since the finite difference solution is closer to the solution of the modified equation u t + au x = Du xx (8.5) than the exact solution of the original linear advection equation (8.3), we check the accuracy of the numerical moments based on the solution of the modified equation (8.5). If the upwind scheme is used then D = a x( aλt x) [5, 6]. Thus D x. The exact solution of the equation (8.5) with initial data (8.4) is ũ(x, t) = x at 4Dt π The exact solution of (8.3) with initial data (8.4) at t is 0 e z dz. (8.6) u(x, t) = {, x < at ;, x > at. (8.7) which has the property δ(u(x, t))dx = 0. When evaluating the δ-integration concentrated on (8.6), one gets 4πDt δ(ũ(x, t))dx = ũ (at, t) = x at = ( ) e 4Dt x=at 4πDt. Since D x, the accuracy of numerical δ-integral based on (8.6) is only halfth order. This implies that the evaluation of δ-integration in (.8)(.9) based on the finite difference solution for (8.3) is also only halfth order. 40